Can I synchronize CC's Read and Date Read information with calibre?

Yes, as long as you are using calibre 1.28 or later. CC's Read information can be synchronized with a calibre Yes/No custom column, and CC's Date Read information can be synchronized with a calibre Date custom column. Syncing happens when you connect to calibre as a wireless device.

You set this up using the following steps:

1. Create the Yes/No column in calibre. To do this, in calibre click on "Preferences" then "Add your own columns". Click the green Plus to add a new column. Click the "Yes/No" link under Quick Create. Fill out the fields as shown in this screen capture:
   
   You can choose whatever lookup name and column heading you want.
   
   Make a note of the lookup name. You will need it in a later step.
2. Create the Date column in calibre. You do this in a way similar to that used when creating the Yes/No column. After clicking on the green Plus, set the column type to Date. Enter the text for the lookup name and column heading. As before, you can use choose different values for these two values.
   
   Again, make a note of the lookup name.
3. Enter the column lookup names in CC's settings
   In CC, ensure you are not connected then go to Settings and scroll down until you find the "Marking Books Read" options. Tap the setting to open the following sub-dialog.
   
   Tap "Sync Column for Read". A dialog box will open where you choose the calibre Read Yes/No custom column.
   
   Choose the correct column and press OK.
   
   If you wish to sync the date read information, do the same thing for that column by tapping "Sync Column for Date Read" and choosing the corrent calibre Date custom column.